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636 Brazilian Journal of Physics,vol.35,no.3A,September,2005Effects of Torsion on Electromagnetic FieldsLa´ercio DiasLaborat´orio de F´sica Te´orica e Computacional,Departamento de F´sica,Universidade Federal de Pernambuco,50670-901,Recife,PE,Braziland Fernando MoraesLaborat´orio de F´sica Te´orica e Computacional,Departamento de F´sica,Universidade Federal de Pernambuco,50670-901,Recife,PE,Brazil andDepartamento de F´sica,CCEN,Universidade Federal da Para´ba,Cidade Universit´aria,58051-970 Joao Pessoa,PB,BrazilReceived on 14 February,2005In this work,we investigate the effects of torsion on electromagnetic elds.As a model spacetime,en-dowed with both curvature and torsion,we choose a generalization of the cosmic string,the cosmic dislocation.Maxwell's equations in the spacetime of a cosmic dislocation are then solved,considering both the case of astatic,uniform,charge distribution along the string,and the case of a constant current owing through the string.We nd that the torsion associated to the defect affects only the magnetic eld whereas curvature affects bothelectric and magnetic elds.Moreover,the magnetic eld is found to spiral up around the defect axis.I.INTRODUCTIONThe study of electromagnetism in a curved backgroundhas very important astrophysical implications as for exam-ple helping the understanding of the signals received fromneutron stars and maybe also from black holes.Electromag-netic processes near such objects certainly will have general-relativistic effects.After the generalization of Einstein's grav-itational theory to include torsion,done by Hehl and cowork-ers [1],one might ask what are the effects of torsion on elec-tromagnetic elds.In this work we study a simple but illus-trative case:the electromagnetic eld produced by a cyllindri-cally symmetric source coincident with a topological line de-fect that carries both curvature and torsion.Topological structures like domain walls,strings andmonopoles may have been produced by phase transitions in-volving spontaneous symmetry breaking in the early uni-verse [2].Such defects are associated to curvature singu-larities [3] and are solutions to Einstein's eld equations.Although astronomical observations keep indicating that themacroscopic geometry of the universe is Riemannian it ispossible that torsion may appear near curvature singularities[4].Line defects containing torsion,like dislocations,appearwithin Einstein-Cartan-Sciama-Kibble gravitation theory [1]in Riemann-Cartan spacetime U4.We consider the cosmic dislocation [5] spacetime whosemetric is given byds2=¡dt2+dr2+2r2d2+(dz + d )2;(1)in cylindrical coordinates.The parameter  is associated withthe angular decit of a cosmic string spacetime.The valuesof  are restricted to the interval 0 < <1,since the lineardensity of mass of a cosmic string,given by µ =(1¡ )=4G,must be positive.The parameter  is related to the torsionassociated to the defect.For dislocations in solid state physics is related to the Burgers vector~b by  =b2.This topological defect carries both torsion and curvature,both appearing as conical singularities on the z-axis.The onlynonzero component of the torsion tensor in this case is givenby the two-form[6]Tz=22(r)dr ^d;(2)where 2(r) is the two-dimensional delta function.Analo-gously,the nonvanishing components of the curvature two-formare [6]Rr=¡Rr=2(1¡ )2(r)dr ^d:(3)This study intends to show how the changes introduced inthe geometrical structure of spacetime by a topological linedefect affect the solutions of Maxwell's equations.We areinterested in the following two cases involving a cosmic dis-location:(i) the defect carries a density of charge and (ii) itcarries a current.Asimilar problemwas handled by M.F.A.daSilva et al.[7,8],who calculated the magnetostatic eld dueto an electric current placed in the gravitational backgroundof a rotating cosmic string.In their work,torsion comes fromrotation,thus coupling time to the angular coordinate .Here,torsion comes fromthe coupling between  and z,as it is clearfrom Eq.(1).Other cases,with spherical symmetry,haveappeared in the literature.For example,the electrostatic eldand the potential of a point charge in the Schwartzschild met-ric obtained by Linet [9] and the magnetostatic eld of a loopcurrent around a black hole obtained by Petterson [10].Re-lated to these problems is the question of the self-force onelectric and magnetic sources in the presence of topologicaldefects,the object of much attention in recent years [11]-[19].We restrict ourselves to the study of the generation of elec-tric and magnetic elds by static sources.To facilitate thecalculations,we consider the approximation [10] where theelectromagnetic eld is taken as a weak perturbation on thespacetime metric.Thus the inuence of the metric on theelectromagnetic eld is much stronger than the inuence ofthe electromagnetic eld on the metric.In this approxima-tion,the task of solving Einstein-Maxwell equations reducesto solving Maxwell equations in covariant form.La´ercio Dias and Fernando Moraes 637This work is organized as follows.In Section II,we deriveMaxwell equations in the cosmic dislocation spacetime.InSection III we solve them for the electrostatic eld generatedby a line of charge.We nd that there is no effect of torsionon the electric eld.On the other hand curvature ampliesit.In Section IV we calculate the magnetostatic eld of a linecurrent in the spacetime of the cosmic dislocation,nding aninteresting effect:the peculiarities of the metric give rise to az-component of the eld.Finally,in Section V we we presentour concluding remarks.We observe that in this paper we usegeometrical units.II.MAXWELL EQUATIONS IN THE COSMICDISLOCATION SPACETIMEWe start by writting Maxwell equations using differentialforms:dF =0 (4)and?d?F =J;(5)whereF =12Fµdxµ^dx=B+E^dt (6)is the Faraday two-formand J is the current density one-formgiven byJ =¡ dt +Jrdr +Jd +Jzdz:(7)In Eq.(6) the magnetic eld is represented by the two-form Band the electric eld by the one-form E.The Faraday two-form in terms of its components is there-foreF =F zd ^dz +Fzrdz ^dr +Frdr ^d+Frtdr ^dr +F td ^dt +Fztdz ^dt:(8)Eqs.(6) and (8) imply thatB =F zd ^dz +Fzrdz ^dr +Frdr ^d (9)andE^dt =Frtdr ^dt +F td ^dt +Fztdz ^dt;(10)which leads toEr=FrtE=F tEz=Fzt;(11)whereE ´Erdr +Ed +Ezdz:(12)We observe that the electric eld vector components(Er;E;Ez) are related to the one-form E components(Er;E;Ez) by the metric in the usual way contravariant andcovariant vector components are related.In the same way,themagnetic eld vector components (Br;B;Bz) are related to amagnetic eld one-form B1components (Br;B;Bz).Never-theless,the two-form B is related to B1by the Hodge?oper-ation:?B =B1^dt:(13)Applying the Hodge?operator on Eq.(9) we obtain?B =µ2r2+2 rFzr+ rFr¶d ^dt+µ1 rFr+ rFzr¶dz ^dt +F z rdr ^dt:(14)Therefore,we identify the components of the magnetic eldone-formB1Br=1 rF zB=2r2+2 rFzr+ rFr(15)Bz=1 rFr+ rFzr:Now,applying the Hodge?operator on Eq.(10) we obtain?(E^dt) = ¡  rFrtd ^dz¡µ1 rF t¡ rFzt¶dz ^dr (16)¡µ2r2+2 rFzt¡ rF t¶dr ^d:Finally,using Eqs.(6),(11),(14),(15) and (16),we obtain?d?F =f1 r Bz¡1 r B z¡ Er tgdr+ f r B r¡ r Br+2r2+2 r Br z¡2r2+2 r Bz r¡ E tgd+ f1 r B r¡1 r Br+ r Br z¡ r Bz r¡ Ez tgdz¡ f1r r(rEr) +(12r2E¡2r2Ez)+12r2 z[(2r2+2)Ez¡ E]gdt:(17)Care should be taken in interpreting the contravariant com-ponents of the elds since the metric (1) is associated to a non-orthonormal basis (~et;~er;~e;~ez),where gµ=~eµ¢~e.There-fore we need to relate the components of the electric and mag-netic eld one-forms to the respective vectors in a normalizedbasis,such that in the no defect limit we recover the eldsgenerated by a line source in at spacetime.The new basis(~et;~er;~e;~ez) is simply obtained by~eµ=~eµpgµµ:(18)638 Brazilian Journal of Physics,vol.35,no.3A,September,2005The components of a generic 1-form A ´ Ardr +Ad +Azdz are related to the components of the equivalent vector~A=Ar~er+A~e+Az~ez,expressed in the normalized (but non-orthogornal) basis,by:Ar=Ar(19)A=q2r2+2A+ Az(20)Az=p2r2+2A+Az:(21)After some algebraic manipulations we nally obtain Eq.(5) in terms of the components of the electric,magnetic andcurrent density vectors1r r(rEr) +1p2r2+2 E+ Ez z=;(22)1 rÃp2r2+2¡q2r2+2 z!B+1 rµ¡ z¶Bz=Jr+ Er t;(23)p(2r2+2) r" Br z¡ rÃBz+p2r2+2B!#=J+ E t;(24)1 r· rµq2r2+2B+ Bz¶¡ Br¸=Jz+ Ez t:(25)Notice that Eq.(22) corresponds to Gauss lawand that Eqs.(23 - 25) correspond to Ampere-Maxwell law.In a similar way,Eq.(4) leads to1r r(rBr) +1p2r2+2 B+ Bz z=0;(26)1 rÃp2r2+2¡q2r2+2 z!E+1 rµ¡ z¶Ez+ Br t=0;(27)p2r2+2 r" Er z¡ rÃEz+p2r2+2E!#+ B t=0;(28)1 r· rµq2r2+2E+ Ez¶¡ Er¸+ Bz t=0:(29)Now,Eq.(26) describes the absence of magnetic monopolesand Eqs.(27 - 29) correspond to Faraday law.III.ELECTRIC FIELD OF THE LINE CHARGEIn this section we briey discuss the case of a uniform lineof charge coincident with the cosmic dislocation.In this case,the charge density is described by (r) =2 (r)r;(30)where  is the linear charge density.The presence of  in thisexpression is due to the change in the volume element causedby the string metric.The symmetries of the problem suggest that Er=Er(r);E=E(r) and Ez=Ez(r).Eqs.(27-29) imply readilythatE(r) =Ez(r) =0 (31)and Eq.(22) givesEr(r) =21r:(32)This result might be explained by a simple argument basedon the electric eld lines,as follows.The process of creat-ing the defect involves cutting out a wedge of space,whichleaves less volume for the eld lines to spread through.Thisincreases the density of eld lines therefore corresponding toan amplication of the electric eld amplitude.This shouldbe compared to the amplication found in the magnetostaticeld of a current-carrying cosmic string [7]).IV.MAGNETIC FIELD OF THE LINE CURRENTNow we treat the case where a current ows along the de-fect.The important equations now are (23 - 25).Here thesymmetry suggests that the nonvanishing components of themagnetic eld are B= B(r) and Bz= Bz(r).With this,inthe region r >0,Eqs.(24) and (25) turn intoddrÃBz+p2r2+2B!=0 (33)ddrµq2r2+2B+ Bz¶=0:(34)We have thus a coupled set of equations of very simple so-lution:B(r) =k1p2r2+22r2(35)andBz(r) =k2¡22r2;(36)where k1and k2are integration constants.In order to deter-mine these constants we withdraw the defect by setting  =1La´ercio Dias and Fernando Moraes 639FIG.1: -component of the magnetic eld:¤in at spacetime (  =1 and  =0),± in the cosmic string spacetime ( =0:5 and  =0),¦ in the cosmic dislocation spacetime ( =0:5 and  =1)and  =0.Thus,we recover the magnetic eld of a line cur-rent in at spacetime:B =1; =0(r) =I2 r;(37)Bz =1; =0(r) =0;(38)where I is the electric current.Hence,we have k1=I2andk2= 0.Substituting this into Eqs.(35) and (36),we nallygetB(r) =I2p2r2+22r2;(39)Bz(r) =¡I22r2:(40)The coupling between the angular and the z coordinatesbrings about an unexpected component of the magnetic eld,which vanishes properly in the no torsion limit !0.In Fig.1 it is shown the  -component of the magnetic eld in a fewillustrative cases.In what follows we take a closer look at themagnetic eld lines in this torsioned space.In order to nd the integral curves (magnetic eld lines) ofour vector (magnetic) eld we need to solve the parametricsystembelow:r(t) = Br(r;;z) (t) = B(r;;z) (41)z(t) = Bz(r;;z);where t is a parameter.Since Br=0,B=Bpgand Bz=Bz(see Eq.(18)) and with Eqs.(39) and (40) we haver(t) = 0 (t) =I212r2(42)z(t) = ¡I22r2;whose solution isr(t) = r0 (t) =I22r20t +0(43)z(t) = ¡I22r20t +z0;where 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